Properties

Label 6012.77
Modulus $6012$
Conductor $1503$
Order $498$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6012, base_ring=CyclotomicField(498))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,415,438]))
 
pari: [g,chi] = znchar(Mod(77,6012))
 

Basic properties

Modulus: \(6012\)
Conductor: \(1503\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(498\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1503}(77,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6012.bb

\(\chi_{6012}(29,\cdot)\) \(\chi_{6012}(65,\cdot)\) \(\chi_{6012}(77,\cdot)\) \(\chi_{6012}(137,\cdot)\) \(\chi_{6012}(173,\cdot)\) \(\chi_{6012}(185,\cdot)\) \(\chi_{6012}(209,\cdot)\) \(\chi_{6012}(221,\cdot)\) \(\chi_{6012}(281,\cdot)\) \(\chi_{6012}(293,\cdot)\) \(\chi_{6012}(317,\cdot)\) \(\chi_{6012}(329,\cdot)\) \(\chi_{6012}(353,\cdot)\) \(\chi_{6012}(365,\cdot)\) \(\chi_{6012}(461,\cdot)\) \(\chi_{6012}(509,\cdot)\) \(\chi_{6012}(533,\cdot)\) \(\chi_{6012}(545,\cdot)\) \(\chi_{6012}(617,\cdot)\) \(\chi_{6012}(653,\cdot)\) \(\chi_{6012}(677,\cdot)\) \(\chi_{6012}(689,\cdot)\) \(\chi_{6012}(725,\cdot)\) \(\chi_{6012}(749,\cdot)\) \(\chi_{6012}(761,\cdot)\) \(\chi_{6012}(857,\cdot)\) \(\chi_{6012}(893,\cdot)\) \(\chi_{6012}(929,\cdot)\) \(\chi_{6012}(965,\cdot)\) \(\chi_{6012}(1013,\cdot)\) ...

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{249})$
Fixed field: Number field defined by a degree 498 polynomial (not computed)

Values on generators

\((3007,3341,4681)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{73}{83}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 6012 }(77, a) \) \(-1\)\(1\)\(e\left(\frac{23}{498}\right)\)\(e\left(\frac{29}{249}\right)\)\(e\left(\frac{229}{498}\right)\)\(e\left(\frac{64}{249}\right)\)\(e\left(\frac{19}{166}\right)\)\(e\left(\frac{1}{83}\right)\)\(e\left(\frac{119}{498}\right)\)\(e\left(\frac{23}{249}\right)\)\(e\left(\frac{379}{498}\right)\)\(e\left(\frac{205}{249}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6012 }(77,a) \;\) at \(\;a = \) e.g. 2